Tree-width, clique-width and fly-automata Bruno Courcelle Bordeaux University, LaBRI (CNRS laboratory) References : B.C, Irène Durand: Automata for the verification of monadic second- order graph properties, J. Applied Logic 10 (2012) 368-409 B.C.: From tree-decompositions to clique-width terms, Discrete Applied Maths , 2017, in press (on line on ScienceDirect.com). B.C.: Fly-automata for checking MSO2 graph properties, Discrete Applied Maths , 2017, in press (on line on ScienceDirect.com). 1 Topics Fixed-parameter tractable (FPT) graph algorithms for monadic second-order (MSO) expressible problems, for graphs of bounded tree-width (twd ) or clique-width (cwd ), based on automata running on algebraic terms denoting the (decomposed) input graphs. Can compute values, not only True / False answers. Tools: Fly-automata (FA): they compute their transitions, to overcome the “huge size problem”, Tree-decompositions encoded by clique-width terms, Linear bounds on cwd in terms of twd for sparse graphs. 2 The basic theorem : Each MSO property of graphs of cwd or twd at most k is decidable in time f(k) x number of vertices . Facts: Extends to MSO properties expressed with edge set quantifications , for graphs of bounded tree-width (not bounded cwd). Graphs given with relevant decompositions, of “small width”. Optimal decompositions are difficult to construct (NP-complete problems). But optimality is not essential. 3 Computation of graph evaluations P(X) is a property of tuples X of sets of vertices (usually MSO expressible) . ∃ X.P( X) : the basic, “Boolean evaluation”. # X.P(X) : number of satisfying tuples X. Sp X.P(X) : spectrum = the set of tuples of cardinalities of the components of the tuples X that satisfy P( X). MinCard X.P(X) : minimum cardinality of X satisfying P(X). 4 Informal review of definitions and basic facts. 1) Graphs are finite, simple, loop-free, directed or not. A graph G can be given by the logical structure ( V G , edg G(.,.) ) = (vertices, adjacency relation) 2) Monadic second-order (MSO) formulas can express p-colorability (and variants), transitive closure, properties of paths, connectedness, planarity (via Kuratowski), etc… 5 Examples : 3-colorability : ∃X ,Y ( X ∩ Y = ∅ ∧ ∀u,v { edg(u,v) ⇒ [(u ∈ X ⇒ v ∉ X) ∧ (u ∈ Y ⇒ v ∉ Y) ∧ (u ∉ X ∪ Y ⇒ v ∈ X ∪ Y) ] } ) The graph is not connected : ∃Z ( ∃x ∈ Z ∧ ∃y ∉ Z ∧ (∀u,v (u ∈ Z ∧ edg(u,v) ⇒ v ∈ Z) ) Planarity is MSO-expressible (no minor K 5 or K 3,3 ). 6 3) Alternative description of graphs : Inc (G) := ( V G U E G , inc G(.,.) ) = (vertices and edges , incidence relation) the bipartite incidence graph of G. MSO formulas on Inc (G) can use quantifications on sets of edges of the considered graph G. Expressing Hamiltonicity of G is possible by an MSO formula on Inc (G) but not on G (edge set quantifications are needed). 7 4) Tree-width ( twd (G) ) is well-known. width of decomposition : 3 dotted lines : equal vertices 8 5) Clique-width : algebraic construction of graphs Vertices are labelled by a,b,c, ... A vertex labelled by a is an a-vertex . Binary operation : disjoint union : ⊕ Unary operations : edge addition denoted by Add a,b Add a,b (G) is G augmented with (un) directed edges from (between) every a-vertex to (and) every b-vertex. vertex relabellings : Relab a b(G) is G with every a-vertex is made into a b-vertex Basic graphs : a denotes a vertex labelled by a 9 The clique-width of G, denoted by cwd (G), is the smallest k such that G is defined by a term using k labels. Such a term is a decomposition of G as a gluing of complete bipartite graphs. k indicates the “complexity of gluings”, not size of components. Classes of bounded clique-width: cographs, cliques, complete bipartite graphs, trees, any class of bounded tree-width. Classes of unbounded clique-width: Planar graphs, chordal graphs. 10 Example 1 : Cliques (a-labelled) have clique-width 2 and unbounded tree-width. Kn is defined by tn where t1 = a tn+1 = Relab b a( Add a,b (t n ⊕ b) ) Example 2 : Cographs (a-labelled) are generated by ⊕ and ⊗ defined by: G ⊗ H = Relab b a ( Add a,b ( G ⊕ Relab a b(H) ) ) = G ⊕ H with “all edges” between G and H. 11 Remark : An algebraic expression of tree-width is possible, by using parallel composition G // H instead of disjoint union G ⊕ H. This operation glues G and H by fusing, for each label a, the (unique ) a-vertex of G and the ( unique ) a-vertex of H. But the construction of an automaton running on terms over // denoting graphs G of twd < k intended to check an MSO property of Inc (G) is more complicated because of these fusions. The basic fact for ⊕ is : G ⊕ H = ϕ (X) if and only if G = ψ 1(X ∩ VG) and H = θ 1(X ∩ VH) or G = ψ 2(X ∩ VG) and H = θ 2(X ∩ VH) … or G = ψ p(X ∩ VG) and H = θ p(X ∩ VH) 12 Comparing tree-width and clique-width (undirected graphs) twd(G) - 1 cwd (G) < 3. 2 (Corneil & Rotics, the exponential is not avoidable) If a box of the tree-dec has k vertices, then 2 k-1 labels may be necessary to specify how the vertices below it are linked to its vertices. 13 For which classes do we have cwd(G) = O(twd(G) c ) for fixed c, and with “good values” of c and of hidden constants ? Graph class cwd (G) where k = twd (G) planar 6k – 9 ( 32k – 57 if directed) degree < d k.d + 1 incidence graph k + 3 ( 2k + 4 if directed) 1-planar 18k - 29 p-planar O(k) ? q at most q. n edges for n vertices O(k ) where q << k These results hold for directed graphs. 14 Remark : About incidence graphs of graphs of bounded tree-width and MSO2 properties. MSO2 means expressed by an MSO formula using edge set quantifications. Example : There exists a set of edges forming a perfect matching, or forming a Hamiltonian path. Not possible without such quantifications. 1) From of a tree-decomposition of G of width k, we construct a clique-width term t for Inc(G) of “small” width k+3 (or 2k+4 ); no exp. ! 2) We translate an MSO2 formula ϕ for G into an MSO formula θ for Inc(G) . 3) The corresponding automaton A( θ) takes term t as input. More remarks to come. 15 Proof method for making tree-decompositions into cwd terms For a graph G and Y a set of vertices : µ ∩ ∉ G(Y) := the number of sets N G(x) Y for x Y. (N G(x) : neighbours of x) More generally, neighbourhood complexity : r r µ G(Y) := the number of sets N G(x) ∩ Y for x ∉ Y. r (N G(x) : neighbours at distance at most r of x) 16 Lemma : If twd(G) < k, and µG(Y) < m whenever Y < k + 1 , then cwd(G) < m + 1. For each graph class, we bound µG(Y) in terms of Y . For planar graphs, we use the bound 3n - 6 on the number of edges ; for q-sparse graphs, we use an orientation of indegree at most q. In all cases we transform a tree-decomposition into a clique-width term based on the same tree. 17 Proof sketch for planar graphs. Enough to consider a bipartite graph with vertex set X U Y and Y = k. There are at most k+1 sets NG(x) ∩ Y for x of degree 0 or 1, (x ∈ X). There are at most 3k-6 sets NG(x) ∩ Y for x of degree 2 : each of them corresponds to an edge of a planar graph with vertex set Y. There are at most 2k-4 vertices x of degree > 2 : let Z be these vertices : 3. Z < E < 2.( Z + k ) - 4 (planar bipartite). Total : k+1 + 3k-6 + 2k -4 = 6k - 9. 18 Graph classes of bounded expansion (Nesetril, Ossona de Mendez) Some cases : bounded degree, minor closed, hence planar or bounded tree-width, topologically closed (by contracting paths ), p-planar, k-colorable. Definition : A class C has bounded expansion if ∀ d ∃ c ∀ G ∈ C and H a d-shallow minor of G, we have EH < c . VH d-shallow minor : contracting connected subgraphs of radius < d 19 Theorem (Reidl et al.) : A class C has bounded expansion iff for each r , we have : r ∃ c ∀ G ∈ C ∀ Y ⊆ VG : µ G(Y) < c. Y Hence, if C has bounded expansion (take r = 1) : cwd (G) = O( twd (G) ) for all G ∈ C Whence, the answer for p-planar graphs. 20 Discussion : - the constants are “bad” (exponential); - however, they are not reached , or in weird cases only ; - better bounds (cf. above for planar and 1-planar graphs) should be determined for classes of particular interest ; - the algorithm given below works for arbitrary tree-decompositions (given by normal trees ) as input; the time and space need not be huge (no “search”). 21 Remark : For a graph of tree-width d, given by a non-optimal tree-decomposition of width k, we obtain a clique-width term of width at most f(d). k for some fixed function f . d/2 The results of Corneil and Rotics give f(d) > 2 / d. To be done : “good” estimation of f(d). 22 More on sparse graphs : nowhere dense graph classes Include : bounded expansion and locally bounded expansion and locally bounded tree-width. Definition (Nesetril, Ossona de Mendez) : A class C is nowhere dense if ∀ d ∃ c ∀ G ∈ C ∀ H d-shallow minor of G : ω(H) < c (ω(H) := max size of a clique in H).